Problem: Let $S$ be the set of all nonzero real numbers.  Let $f : S \to S$ be a function such that
\[f(x) + f(y) = f(xyf(x +  y))\]for all $x,$ $y \in S$ such that $x + y \neq 0.$

Let $n$ be the number of possible values of $f(4),$ and let $s$ be the sum of all possible values of $f(4).$  Find $n \times s.$
Fix $s \in S.$  Setting $y = s - x,$ we get
\[f(x) + f(s - x) = f(x(s - x)f(s)). \quad (*)\]This holds for all $x \in S,$ $x \neq s.$

Consider the equation
\[s - x = x(s - x) f(s).\]The solutions in $x$ are $x = s$ and $x = \frac{1}{f(s)}.$  Since $x \in S,$ $f(s)$ is well-defined.  Furthermore, $f(s) \neq 0,$ so $\frac{1}{f(s)}$ is well-defined.  If $f(s) \neq \frac{1}{s},$ then we can set $x = \frac{1}{f(s)}$ in $(*),$ which gives us
\[f \left( \frac{1}{f(s)} \right) + f \left( s - \frac{1}{f(s)} \right) = f \left( s - \frac{1}{f(s)} \right).\]Then $f \left( \frac{1}{f(s)} \right) = 0,$ contradiction.

The only possibility then is that $f(s) = \frac{1}{s}.$  In other words,
\[f(x) = \frac{1}{x}\]for all $x \in S.$

We can check that $f(x) = \frac{1}{x}$ works, so $n = 1$ and $s = \frac{1}{4},$ so $n \times s = \boxed{\frac{1}{4}}.$